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Description

Theory of Elastic Thin Shells discusses the mathematical foundations of shell theory and the approximate methods of solution. The present volume was originally published in Russian in 1953, and remains the only text which formulates as completely as possible the different sets of basic equations and various approximate methods of shell analysis emphasizing asymptotic integration.
The book is organized into five parts. Part I presents the general formulation and equations of the theory of shells, which are based on the well-known hypothesis of the preservation of the normal element. Part II is devoted to the membrane theory--the most widely used approximate method of analysis of shells that was formulated at approximately the same time as the more general bending theory. In Part III methods of analysis of circular cylindrical shells with the aid of trigonometric series are considered. Part IV is essentially mathematical in character and its purpose is to justify the approximate methods of shell analysis. In Part V approximate methods of analysis of shells are formulated.

Table of Contents

Translation Editor's Preface to English Edition
Author's Preface to English Edition
Preface
Part I Basic Relations in the Theory of Shells
Chapter 1. Brief Outline of the Theory of Surfaces
1. Curvilinear Coordinates on a Surface and the First Quadratic Form
2. Basic and Auxiliary Trihedra of a Surface. Decomposition of an Arbitrary Vector Along Axes of Basic and Auxiliary Trihedra
3. Gauss-Weingarten Derivative Formulas. Codazzi-Gauss Equations
4. Resolution of Derivatives of an Arbitrary Vector along the Axes of the Basic and Auxiliary Trihedra
5. Second Quadratic Form of a Surface and Dupin's Indicatrix
6. Conjugate Lines, Lines of Curvature, Asymptotic Lines
7. Gaussian Curvature and Bending of Surfaces
8. Fundamental Formulas of the Theory of Surfaces in Orthogonal Coordinates
Chapter 2. Static and Geometric Relations of the Theory of Shells
9. Forces and Moments
10. Forces and Moments along Oblique Sections
11. External Loads
12. Equilibrium Equations of the Shell
13. Stress Functions
14. Vectors of Elastic Displacement and of Elastic Rotation of the Middle Surface
15· Components of Tangential Deformation (Strain) of the Middle Surface of the Shell
16. Expansions for the Derivatives of the Vector of Elastic Displacement
17· Components of Bending Deformation (Strain) of the Middle Surface
18. Expressions for Derivatives of the Vector of Elastic Rotation
19· Expressions for Components of Deformation and Angles of Rotation in Terms of Displacements
20· Determination of Displacements on the Basis of Given Components of Deformation. Equations of Compatibility of Strain
21. Transformation of Components of Strain
Chapter 5. Relations of Elasticity. General Theorems of the Theory of Shells
22. Fundamental Hypothesis of the Theory of Shells
23. Relations of Elasticity
24. Supplementary Equations of the Theory of Shells
25. Work of Forces and Moments, of Thin Shells
26. Strain Energy
27· Analysis of Some Variants of Elasticity Relations
Chapter 4. Fundamental Equations of the Theory of Shells
28. Summary of Fundamental Relations of the Theory of Shells
29· Complete System of Equations of the Theory of Shells
30. Static-Geometric Analogy
31. Equations of Compatibility in Terms of Forces and Moments
32. Equations of Equilibrium in Terms of Displacements
33· Boundary Conditions
Part II Membrane Theory
Chapter 5· Membrane Theory of Shells of Arbitrary Shape
1. General Assumptions of Membrane Theory
2. Static, Geometric and Mixed Problems of Membrane Theory
3. Boundary Conditions in Membrane Theory
4· Three Classes of Membrane Shells
5· Relations between Membrane Theory and the Theory of Infinitesimal Flexures of Surfaces
6. Conjugate Geometric and Static Problems of Membrane Theory
7· Membrane Shell of Positive Curvature with One Geometric Condition
Chapter 6. Membrane Theory of Shells of Zero Curvature
8. Curvilinear Coordinates on Cylindrical and Conical Surfaces
9. General Integral of Equations of Membrane Theory of Shells of Zero Curvature
10. Boundary Conditions
11. Examination of the State of Stress of a Cylindrical Membrane Shell
12. Examples of Analysis of Cylindrical Membrane Shells
13. Examples of Analysis of Cylindrical Membrane Shells, Continued
Chapter 7. Membrane Theory of Spherical Shells
14. Transformation of Membrane Equations of a Spherical Shell
15. Integration Methods for the Equations of Membrane Theory of Spherical Shells
16. Application of the Methods of the Theory of Functions of a Complex Variable to the Analysis of Spherical Membrane Shells
17. Integral Equations of Equilibrium
18. Static Meaning of Poles of the Complex Stress Function
Chapter 8. Analysis of Closed Spherical Membrane Shells
19. Analysis of Closed Spherical Membrane Shells under the Action of Concentrated. Forces and Moments
20. Example
21. Displacements of a Closed Spherical Shell Subjected to Concentrated forces and Moments
22. Analysis of Closed Spherical Membrane Shells Subjected to Distributed Loads
23. Generalizations
Chapter 9. Analysis of Membrane Shells Taking Boundary Conditions into Account
24. The Simplest Problems in Which Account Must be Taken of Boundary Conditions
25. Examples
26. Number of Solutions of Static and Geometric Problems for Membrane Shells of Positive Curvature
27. Examples of Statically Determinate and Geometrically Variable Membrane Shells
Part III Circular Cylindrical Shells
Chapter 10. Method of Expansion in Trigonometric Series
1. Basic Equations of the Theory of Cylindrical Shells
2. The Solving Equation of Circular Cylindrical Shells
3. Application of Trigonometric Series to the Analysis of Circular Cylindrical Shells
Chapter 11. Analysis of Closed Cylindrical Shells
4. Basic Formulas for Analysis
5. Properties of Roots of the Characteristic Equation. Simplification of the Characteristic Equation
6. Physical Meaning of Zero Roots of the Characteristic Equation
7. Analysis of the State of Stress of Closed Cylindrical Shells
8. Approximate Methods of Analysis of the Basic State of Stress of Circular Cylindrical Shells
9. Approximate Methods of Analysis of Edge Effects
10. States of Stress Corresponding to Large Values of m
11. Imposition of Boundary Conditions
Chapter 12. Analysis of Open Cylindrical Shells
12. Basic Formulas for Analysis
13. Properties of Roots of Characteristic Equation
14. Analysis of the State of Stress in Open Cylindrical Shells
15. Approximate Methods of Analysis of Open Cylindrical Shells
16. Imposition of Boundary Conditions
Part IV Analysis of the State of Stress in an Arbitrary Shell
Chapter 15. Asymptotic Integrations of Partial Differential Equations
1. Classification of Linear Differential Operators with Partial Derivatives
2. Nomenclature and Notations
3. Asymptotic Expansion of Integrals of a Homogeneous Differential Equation
4. Three Fundamental Cases
5. Construction of Functions of Variation
6. Integrals with Given Non-Characteristic Supporting Contour
7. Case of Multiple Characteristics
8. Integrals with Given Characteristic Supporting Contour
9. Asymptotic Expansion of Particular Solution of a Nonhomogeneous Partial Differential Equation
10. Example
Chapter 14. Asymptotic Integration of Equations of the Theory of Shells
11. Asymptotic Integration of a System of Equations
12. Non-Contradictory values of Indices of Intensity
13. Construction of Functions of Variation
14. Determination of Coefficients of Asymptotic Expansion of Functions of Intensity for Fundamental Integrals
15. Construction of Approximate Equations of the Theory of Shells
16. Asymptotic Error of Equations of Membrane Theory
17· Elementary States of Stress in an Arbitrary Shell
18. The Complete State of Stress in an Arbitrary Shell
Chapter 15. Elementary States of Stress
19. Fundamental State of Stress. Membrane and Pure Bending States of Stress
20. Approximate Equations for States of Stress with Large Indices of Variation
21. Region of Applicability of Equation (20.ll)
22. Simple Edge Effect
23. Integration of the Solving Equation of Simple Edge Effect
24. Solving Equations of Non-degenerate Generalized Edge Effects
25. Solving Equations of Generalized Edge Effect in a Shell of Zero Curvature
26. Range of Applicability of Solving Equations (25.5)
27. Further Simplification of Solving Equations (25.5)
28. Range of Applicability of Membrane Theory in the Analysis of Shells of Zero Curvature
29. Estimating the Accuracy of Construction of a Complete State of Stress
Part V Approximate Methods of Analysis of Shells
Chapter 16. Application of Expansions in Orthogonal Functions to the Analysis of Shells
1. Expansion of Functions in Fourier Series
2. Methods of Construction of Closed Orthogonal Systems of Functions
3. Continuation
4. Index of Variation of the State of Stress and of External Loading
Chapter 17. General Approximate Methods
5. Membrane Theory
6. Region of Applicability of Membrane Theory
7. Properties of the Simple Edge Effect
8. Approximate Theory of the Simple Edge Effect
9. Analysis of Shells by the Membrane Theory with Consideration of Edge Effects
10. Particular Cases
11. Example
12. Approximate Methods of Analysis of Shells with Large Indices of Variation
13. Example
14. Shells with Non-rigidly Supported Edges
Chapter 18. Cylindrical and Conical Shells
15. The Generalized Edge Effect in a Shell of Zero Curvature
16. The Solving Equations of the Generalized Edge Effect in Shells of Zero Curvature
17. Integration of the Solving Equations of the Generalized Edge Effect for Cylindrical Shells
18. Imposition of Boundary Conditions .
19. Integration of Equations of the Generalized Edge Effect for Conical Shells
20. Analysis of the State of Stress of Shells of Zero Curvature
21. Approximate Theory of the Non-degenerate Edge Effect
22. Integration of Equations of the Non-degenerate Edge Effect for Cylindrical and Conical Shells
23. Integration of System (22.9)
24. Continuation
25. Tables of Elastic Reactions and of Elastic Displacements of Cylindrical Shells of Medium Reduced Length
26. Example
27. Analysis of Cylindrical Shells of Medium Reduced Length Subjected to Loads Distributed along a Generator
28. Example
29. Analysis of Conical Shells
Author's Addendum to English Edition (Some Mathematical Problems of the Linear Theory of Elastic Thin Shells)
Author's Amendments
Author Index
Subject Index

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